The QCB upper bounds the error probability, \(P_{e}^{\star }\ge P_{e}\) ,
for both the \(D=2\)  case [1]}, [2]}, [3]}
or the \(D>2\)  cases [4]}. There is no fundamental
difference between these cases, it is only the closest two states
that determine the asymptotic decay of \(P_{e}\)  when \(D>2\) . I will
restrict to \(D=2\)  from hereon to make a correspondence with Touil
et al. [5]}. The error probability (bound)
is
\(P_{e}^{\star }=\min _{0\le c\le 1}p_{1}^{c}p_{2}^{1-c}\prod _{k\in \mathcal {F}}\mathrm {tr}\left[\rho _{\left.k\right|1}^{c}\rho _{\left.k\right|2}^{1-c}\right].\) 
